Timeline for Are hearts of all $t$-structures on smashing triangulated categories closed with respect to coproducts (also)?
Current License: CC BY-SA 4.0
10 events
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| Jul 25, 2019 at 20:34 | comment | added | Clark Barwick | Of course. Thanks! | |
| Jul 25, 2019 at 18:12 | comment | added | Simone Virili | @ClarkBarwick, for such an example consider the dual situation: consider a Grothendieck category $A$ where products are not exact (this happens for several categories of q.coh. sheaves). Then $A$ is the heart of the canonical $t$-structure in its derived category $D(A)$, but it is not closed under products taken in $D(A)$. Now, for your example, pass to the opposite category. | |
| Jul 25, 2019 at 16:39 | comment | added | Clark Barwick | Agreed. I was initially confused by the phrase 'closed under', so I wanted to underscore the distinction, in case that was contributing to your sense of surprise! (Incidentally, do you have an easy example of such a T for which the coproduct (in T) of objects in the heart do not remain in the heart?) | |
| Jul 25, 2019 at 13:24 | comment | added | Mikhail Bondarko | Dear Clark, my problem was that I believed that this (obvious) statement is wrong.:) Besides, in my question I have described the relation between coproducts in the heart and in $T$ precisely; so I hope that this will not cause any confusion. | |
| Jul 25, 2019 at 13:18 | answer | added | Simone Virili | timeline score: 2 | |
| Jul 25, 2019 at 9:04 | comment | added | Clark Barwick | I think one has to be careful about using the word 'closed with respect to coproducts' here. The heart obviously has coproducts, but in the generality you're working, I don't think they are preserved by the inclusion into your triangulated category T. | |
| Jul 25, 2019 at 7:48 | vote | accept | Mikhail Bondarko | ||
| Jul 24, 2019 at 22:05 | answer | added | Jeremy Rickard | timeline score: 5 | |
| Jul 24, 2019 at 20:05 | history | edited | Mikhail Bondarko |
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| Jul 24, 2019 at 8:38 | history | asked | Mikhail Bondarko | CC BY-SA 4.0 |